/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Multi-armed bandits is a rich, multi-disciplinary area that has been studied since 1933, with a surge of activity in the past 10-15 years. This is the first monograph to provide a textbook like treatment of the subject. The work on multi-armed bandits can be partitioned into a dozen or so directions. Each chapter tackles one line of work, providing a self-contained introduction and pointers for further reading. Introduction to Multi-Armed Bandits concentrates on fundamental ideas and elementary, teachable proofs over the strongest possible results. It emphasizes accessibility of the material; while exposure to machine learning and probability/statistics would certainly help, a standard undergraduate course on algorithms should suffice for background. The first four chapters are devoted to IID rewards with adversarial rewards being covered in the next 3 chapters. Contextual bandits are discussed in a separate chapter before the monograph concludes with connections to economics. Each chapter contains a section on bibliographic notes and further directions. Many of the chapters conclude with some exercises. Introduction to Multi-Armed Bandits provides an accessible treatment for students of a topic that has gained importance in the last decade. Lecturers can use it as a text for an introductory course on the subject.

/pdf/introduction-to-multi-armed-bandits-4xr7k57ndi.pdf

Introduction to Multi-Armed Bandits

5分で分かる!? 有名論文ナナメ読み：Jacot, Arthor, Gabriel, Franck and Hongler, Clement : Neural Tangent Kernel : Convergence and Generalization in Neural Networks

Multi-armed bandits a simple but very powerful framework for algorithms that make decisions over time under uncertainty. An enormous body of work has accumulated over the years, covered in several books and surveys. This book provides a more introductory, textbook-like treatment of the subject. Each chapter tackles a particular line of work, providing a self-contained, teachable technical introduction and a brief review of the further developments; many of the chapters conclude with exercises. 
The book is structured as follows. The first four chapters are on IID rewards, from the basic model to impossibility results to Bayesian priors to Lipschitz rewards. The next three chapters cover adversarial rewards, from the full-feedback version to adversarial bandits to extensions with linear rewards and combinatorially structured actions. Chapter 8 is on contextual bandits, a middle ground between IID and adversarial bandits in which the change in reward distributions is completely explained by observable contexts. The last three chapters cover connections to economics, from learning in repeated games to bandits with supply/budget constraints to exploration in the presence of incentives. The appendix provides sufficient background on concentration and KL-divergence. 
The chapters on "bandits with similarity information", "bandits with knapsacks" and "bandits and agents" can also be consumed as standalone surveys on the respective topics.

Machine learning methods strive to acquire a robust model during training that can generalize well to test samples, even under distribution shifts. However, these methods often suffer from a performance drop due to unknown test distributions. Test-time adaptation (TTA), an emerging paradigm, has the potential to adapt a pre-trained model to unlabeled data during testing, before making predictions. Recent progress in this paradigm highlights the significant benefits of utilizing unlabeled data for training self-adapted models prior to inference. In this survey, we divide TTA into several distinct categories, namely, test-time (source-free) domain adaptation, test-time batch adaptation, online test-time adaptation, and test-time prior adaptation. For each category, we provide a comprehensive taxonomy of advanced algorithms, followed by a discussion of different learning scenarios. Furthermore, we analyze relevant applications of TTA and discuss open challenges and promising areas for future research. A comprehensive list of TTA methods can be found at \url{https://github.com/tim-learn/awesome-test-time-adaptation}.

A Comprehensive Survey on Test-Time Adaptation under Distribution Shifts

We consider the Frank-Wolfe (FW) method for constrained convex optimization, and we show that this classical technique can be interpreted from a different perspective: FW emerges as the computation of an equilibrium (saddle point) of a special convex-concave zero sum game. This saddle-point trick relies on the existence of no-regret online learning to both generate a sequence of iterates but also to provide a proof of convergence through vanishing regret. We show that our stated equivalence has several nice properties, as it exhibits a modularity that gives rise to various old and new algorithms. We explore a few such resulting methods, and provide experimental results to demonstrate correctness and efficiency.

/pdf/on-frank-wolfe-and-equilibrium-computation-2kgwm8kdlw.pdf

On Frank-Wolfe and Equilibrium Computation

We consider the problem of minimizing a smooth convex function by reducing the optimization to computing the Nash equilibrium of a particular zero-sum convex-concave game. Zero-sum games can be solved using online learning dynamics, where a classical technique involves simulating two no-regret algorithms that play against each other and, after $T$ rounds, the average iterate is guaranteed to solve the original optimization problem with error decaying as $O(\log T/T)$. In this paper we show that the technique can be enhanced to a rate of $O(1/T^2)$ by extending recent work \cite{RS13,SALS15} that leverages \textit{optimistic learning} to speed up equilibrium computation. The resulting optimization algorithm derived from this analysis coincides \textit{exactly} with the well-known \NA \cite{N83a} method, and indeed the same story allows us to recover several variants of the Nesterov's algorithm via small tweaks. We are also able to establish the accelerated linear rate for a function which is both strongly-convex and smooth. This methodology unifies a number of different iterative optimization methods: we show that the \HB algorithm is precisely the non-optimistic variant of \NA, and recent prior work already established a similar perspective on \FW \cite{AW17,ALLW18}.

Acceleration through Optimistic No-Regret Dynamics

/pdf/acceleration-through-optimistic-no-regret-dynamics-4hazl3mwu5.pdf

We consider the use of no-regret algorithms to compute equilibria for particular classes of convex-concave games. While standard regret bounds would lead to convergence rates on the order of $O(T^{-1/2})$, recent work \citep{RS13,SALS15} has established $O(1/T)$ rates by taking advantage of a particular class of optimistic prediction algorithms. In this work we go further, showing that for a particular class of games one achieves a $O(1/T^2)$ rate, and we show how this applies to the Frank-Wolfe method and recovers a similar bound \citep{D15}. We also show that such no-regret techniques can even achieve a linear rate, $O(\exp(-T))$, for equilibrium computation under additional curvature assumptions.

/pdf/faster-rates-for-convex-concave-games-szog406l3s.pdf

Faster Rates for Convex-Concave Games

Incorporating a so-called "momentum" dynamic in gradient descent methods is widely used in neural net training as it has been broadly observed that, at least empirically, it often leads to significantly faster convergence. At the same time, there are very few theoretical guarantees in the literature to explain this apparent acceleration effect. Even for the classical strongly convex quadratic problems, several existing results only show Polyak's momentum has an accelerated linear rate asymptotically. In this paper, we first revisit the quadratic problems and show a non-asymptotic accelerated linear rate of Polyak's momentum. Then, we provably show that Polyak's momentum achieves acceleration for training a one-layer wide ReLU network and a deep linear network, which are perhaps the two most popular canonical models for studying optimization and deep learning in the literature. Prior work Du at al. 2019 and Wu et al. 2019 showed that using vanilla gradient descent, and with an use of over-parameterization, the error decays as $(1- \Theta(\frac{1}{ \kappa'}))^t$ after $t$ iterations, where $\kappa'$ is the condition number of a Gram Matrix. Our result shows that with the appropriate choice of parameters Polyak's momentum has a rate of $(1-\Theta(\frac{1}{\sqrt{\kappa'}}))^t$. For the deep linear network, prior work Hu et al. 2020 showed that vanilla gradient descent has a rate of $(1-\Theta(\frac{1}{\kappa}))^t$, where $\kappa$ is the condition number of a data matrix. Our result shows an acceleration rate $(1- \Theta(\frac{1}{\sqrt{\kappa}}))^t$ is achievable by Polyak's momentum. All the results in this work are obtained from a modular analysis, which can be of independent interest. This work establishes that momentum does indeed speed up neural net training.

A Modular Analysis of Provable Acceleration via Polyak's Momentum: Training a Wide ReLU Network and a Deep Linear Network

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Papers

On Frank-Wolfe and Equilibrium Computation

Acceleration through Optimistic No-Regret Dynamics

Acceleration through Optimistic No-Regret Dynamics

Faster Rates for Convex-Concave Games

A Modular Analysis of Provable Acceleration via Polyak's Momentum: Training a Wide ReLU Network and a Deep Linear Network